Applied Mathematics/Laplace Transforms

The Laplace transform is an integral transform which is widely used in physics and engineering.

Laplace Transforms involve a technique to change an expression into another form that is easier to work with using an improper integral. We usually introduce Laplace Transforms in the context of differential equations, since we use them a lot to solve some differential equations that can't be solved using other standard techniques. However, Laplace Transforms require only improper integration techniques to use. So you may run across them in first year calculus.

Notation: The Laplace Transform is denoted as .

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace.

Definition edit

For a function  , using Napier's constant   and a complex number  , the Laplace transform   is defined as follows:


The parameter   is a complex number.

  with real numbers   and  .

This   is the Laplace transform of  .

Explanation edit

Here is what is going on.

Examples of Laplace transform edit

Examples of Laplace transform

In the above table,

  1.   and   are constants
  2.   is a natural number
  3.   is the Delta function
  4.   is the Heaviside function

ID Function Time domain
Laplace domain
Region of convergence
for causal systems
1 Ideal delay    
1a Unit impulse      
2 Delayed nth power with frequency shift      
2a nth Power      
2a.1 qth Power      
2a.2 Unit step      
2b Delayed unit step      
2c Ramp      
2d nth Power with frequency shift      
2d.1 Exponential decay      
3 Exponential approach      
4 Sine      
5 Cosine      
6 Hyperbolic sine      
7 Hyperbolic cosine      
8 Exponentially-decaying sine      
9 Exponentially-decaying cosine      
10 nth Root      
11 Natural logarithm      
12 Bessel function
of the first kind, of order n
13 Modified Bessel function
of the first kind, of order n
14 Bessel function
of the second kind, of order 0
15 Modified Bessel function
of the second kind, of order 0
16 Error function      
17 Constant    
Explanatory notes:

  •   represents the Heaviside step function.
  •   represents the Dirac delta function.
  •   represents the Gamma function.
  •   is the Euler-Mascheroni constant.

  •  , a real number, typically represents time,
    although it can represent any independent dimension.
  •   is the complex angular frequency.
  •  ,  ,  ,   and   are real numbers.
  •  is an integer.
  • A causal system is a system where the impulse response h(t) is zero for all time t prior to t = 0. In general, the ROC for causal systems is not the same as the ROC for anticausal systems. See also causality.

Examples edit

1. Calculate   (where   is a constant) using the integral definition.



2. Calculate   using the integral definition.